I think you only need to worry about health insurance for your father if you want an inheritance from him.
Oops, that sounds cold.
But as long as you are living in a first world country (even in the USA), in an emergency he can go to an ER, or you can call an ambulance, and he will be treated first and asked for payment afterwards. (If it's not an emergency, given your father's attitude you probably won't know about it until it has become an emergency.)
If there is an emergency and you are contacted by your father or by a medical provider, you will probably be upset, and worried, and feel harried and hassled and have to rearrange your life at short notice and with considerable difficulty in order to be present and perhaps make decisions. It is at that point which you may be given a piece of paper to sign, quite possibly being told that it is routine, that it is necessary in order for you to be given information about your father's condition or in order to be allowed to see him or in order to be allowed to talk to the doctor or make decisions about what happens to him. DON'T SIGN ANYTHING under those conditions. And apologies for shouting there, but it would be very easy to make a mistake and sign in haste only to repent at leisure when you found that you had assumed responsibility for his finances. Don't sign anything without assurance from a lawyer whom you personally are paying for the advice that it will not make you responsible for your father's medical expenses.
As long as you have not assumed responsibility for his medical expenses, it will be up to the medical provider to retrieve costs from your father. Usually some arrangement can be reached. If not, your father may become bankrupt through medical expenses, but that too can be dealt with.